series and sequence help

sorry for the image but i am having trouble proving the answers i get while doing my first problem

(a) I know the sequence converges to 0 since if you take the limit as n goes to infinity, then a(n) will go to 0 because fractions multiplied by themselves many times that are below 1 go to 0. but how do i prove this answer??

(b) is this geometric?

(c) does it converge because its a constant ratio by geometric series test?

(d) does it converge to 9?

(e) i know that the answer is yes, .999... is equal to 1. but how do i prove that? would i set up a series of some kind??